Heat broiler to high with top rack set just below the heat source. Place poblano pepper on a baking sheet and broil 8 to 10 minutes, turning occasionally to ensure pepper blackens evenly.
Using tongs, transfer hot pepper to a paper bag or lidded container. Let steam 5 minutes. Once pepper is cool enough to handle, peel and discard skin, stem, and seeds. Dice remaining pepper. Heat olive oil in a medium saucepan set over medium-high heat.
Add onion and jalapeño and cook until softened, about 10 minutes. Reduce heat to medium-low and add poblano. Add milk and processed cheese and stir until smooth.
Add pepper Jack, one handful at a time, stirring between additions.
Taste chile con queso and adjust seasoning as necessary. Serve immediately.